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New Perspectives on Polyhedral and their Structures Santiago Alvarez, Jorge Echeverria

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Santiago Alvarez, Jorge Echeverria. New Perspectives on Polyhedral Molecules and their Crystal Structures. Journal of Physical Organic Chemistry, Wiley, 2010, 23 (11), pp.1080. ￿10.1002/poc.1735￿. ￿hal-00589441￿

HAL Id: hal-00589441 https://hal.archives-ouvertes.fr/hal-00589441 Submitted on 29 Apr 2011

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Physical Organic Chemistry

New Perspectives on Polyhedral Molecules and their Crystal Structures

For Peer Review

Journal: Journal of Physical Organic Chemistry

Manuscript ID: POC-09-0305.R1

Wiley - Manuscript type: Research Article

Date Submitted by the 06-Apr-2010 Author:

Complete List of Authors: Alvarez, Santiago; Universitat de Barcelona, Departament de Quimica Inorganica Echeverria, Jorge; Universitat de Barcelona, Departament de Quimica Inorganica

continuous shape measures, stereochemistry, shape maps, Keywords: polyhedranes

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1 2 3 4 5 6 7 8 9 10 New Perspectives on Polyhedral Molecules and their Crystal Structures 11 12 Santiago Alvarez, Jorge Echeverría 13 14 15 Departament de Química Inorgànica and Institut de Química Teòrica i Computacional, 16 Universitat de Barcelona, Martí i Franquès 1-11, 08028 Barcelona (Spain). Fax: +34-93-490 17 18 7725; e-mail: [email protected] 19 20 For Peer Review 21 22 23 Abstract 24 25 26 Relevant families of ideal polyhedra (Platonic, Archimedean, prisms Johnson, and 27 ) are briefly summarized, and an overview of polyhedral and , existing 28 29 or hypothetical, is presented. The assignment of a polyhedral shape to a specific compound with 30 the help of continuous shape measures and derived tools is also briefly discussed, and 31 32 application of shape analysis to cyclic molecules such as cyclobutane, cyclohexane and 33 cyclooctatetraene is presented to illustrate the usefulness of ideal polyhedra in the 34 35 stereochemical description of non-polyhedral molecules. Finally, the presence of latent 36 octahedral in icosahedral polyhedra is used to design new molecules with nested 37 38 shells of the two supposedly incompatible , and to explain the cubic crystal structures 39 40 of icosahedral molecules such as dodecahedrane and . 41 42 43 44 45 46 47 48 49 50 Keywords: Continuous shape measures, stereochemistry, polyhedral molecules, polyhedranes, 51 polyhedrenes. 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 I filled near full with Pease and Water, the iron Pot 6 and laid on the Pease a leaden cover [...] the Pease 7 dilated [... and] what they increased in bulk was [...] 8 9 pressed into the interstices of the Pease, which they 10 adequately filled up, being thereby formed into pretty 11 regular . 12 13 14 Stephen Hales, Vegetable Staticks, 1727. 15 16 17 Introduction 18 19 While the Platonic tetrahedral coordination around a atom, proposed in 1874 by 20 For Peer Review 21 Jacobus Henricus Van 't Hoff and Joseph Achille Le Bel, represented the cornerstone of a new 22 discipline, stereochemistry, molecules with polyhedral skeletons in which a carbon atom 23 24 occupies each , are relatively newcomers to the world of synthesized and well 25 characterized molecules. The oldest member of the family of polyhedranes, , was 26 [1] 27 reported in 1964. A subsequent geometrical analysis of prisms, Platonic and Archimedean 28 polyhedra, pointed to those that could be made of sp3-hybridized carbon atoms.[2] 29 [3] 30 had to wait more than a decade to see the light, and soon after dodecahedrane could be 31 synthesized.[4] Much more recent is the discovery of buckminsterfullerene, of formula C , with 32 60 33 the shape of a truncated .[5] Besides the Platonic solids, there are other sorts of 34 35 semiregular polyhedra that can be made as purely organic molecules. These include 36 Archimedean solids, prisms and some of the 92 Johnson polyhedra.[6] 37 38 39 With the structure of a polyhedral at hand, we must address two relevant 40 stereochemical questions: Which ideal represents best its stereochemistry? How 41 42 similar is the molecular structure to the ideal shape? Avnir and coworkers have proposed that 43 symmetry[7] and shape[8] should be treated as continuous properties and defined continuous 44 45 symmetry measures (CSM) and continuous shape measures (CShM). These parameters allow us 46 to calibrate the deviation of structures from a given symmetry or shape at the same scale, 47 48 independent of their size or number of vertices. Later on, we showed that one can define a 49 minimal distortion interconversion path between two polyhedra in terms of continuous shape 50 51 measures and therefore numerically evaluate not only the deviation of a given structure from a 52 53 particular polyhedron, but also its deviation from the minimal distortion path between two 54 reference polyhedra.[9] 55 56 57 Herein we wish to present an overview of the main polyhedral organic molecules. We will 58 also present examples of minimal distortion paths between a and a polyhedron to 59 60 organize structural and conformational diversity of cyclohexyl and cyclooctatetraene derivatives. Finally, we will discuss the latency of cubic symmetry in icosahedral and

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1 2 3 4 tetrahedral molecules, and on the implications for crystal packing and for the design of metal- 5 organic frameworks. 6 7 8 Shape and Symmetry of Ideal Polyhedra 9 10 Although molecular shape and symmetry are intimately associated, it is important to stress 11 12 here the main differences between these two properties. Let us consider as an example the two 13 Archimedean polyhedra with 24 vertices shown in Figure 1, the truncated and the truncated 14 15 : they both have (i.e., they belong to the Oh symmetry point 16 17 ), but they differ in their number and type of faces. In other words, they have the same 18 symmetry but different shapes. 19 20 For Peer Review 21 In brief, we say that two objects (molecules) have the same shape if they differ only in size, 22 position or orientation in space. Alternatively, we can say that two objects (molecules) have the 23 24 same shape if they can be superimposed by combinations of translations, rotations and isotropic 25 scaling.[10] On the other hand, two objects have the same symmetry if they remain 26 27 indistinguishable after application of the same set of symmetry operations. It follows that two 28 objects with different shapes may have the same symmetry, and we can therefore conclude that 29 30 shape is a more stringent criterion than symmetry, as illustrated by the examples in Figure 1. 31 There are, however some cases in which shape and symmetry are equivalent, and these 32 33 correspond precisely to the Platonic polyhedra, since, e. g., all four vertex polyhedra with 34 35 have the same shape, and the same happens for the octahedron, the cube, 36 the and the icosahedron. Among the Archimedean polyhedra, only for the 37 38 and the are shape and symmetry equivalent. 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Figure 1. Two Archimedean polyhedra with octahedral symmetry and 24 vertices: the truncated 53 cube (left) and the (right). 54 55 56 The set of Platonic solids (, octahedron, cube, icosahedron and dodecahedron) 57 are the most regular polyhedra, each having all its edges, faces and vertices equivalent. Second 58 59 to the Platonic solids in regularity come the Archimedean polyhedra, in which all vertices (but 60 not faces or edges) are equivalent. However, these two families provide us with only a limited number of ideal shapes and we should have at hand other sets of less regular polyhedra. For

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1 2 3 4 instance, the prisms, whose ideal shapes are conventionally those with all edges of the same 5 length. As a result, all the faces of the ideal prisms are regular . If more reference 6 [11] 7 shapes are needed we can make recourse to the 92 Johnson polyhedra, defined as those that 8 have as faces only regular polygons with edges of the same length, excluding the Platonic, 9 10 Archimedean, prismatic and antiprismatic polyhedra. Yet another important family of 11 polyhedra is that of the Fullerenes. Even if the name comes from C , Buckminsterfullerene, 12 60 13 that has the shape of the Archimedean , it refers to all those polyhedra 14 with twelve pentagonal and any number of hexagonal faces,[12] in which all vertices are three- 15 16 connected. 17 18 19 We note that the Platonic and Archimedean polyhedra belong to one of three high- 20 symmetry point groupFors, icosahedral Peer (Ih), octahedral Review (Oh) or tetrahedral (Td), or to their 21 [13] 22 rotational subgroups. In principle, icosahedral and octahedral symmetries are incompatible, 23 since the former features five-fold rotational axes, while the latter has four-fold axes instead. 24 [14] 25 However, we have recently shown that icosahedral polyhedra have latent octahedral 26 symmetry[15] that can be revealed in chemical structures by an appropriate substitution pattern. 27 28 We will go back to some chemical implications of this symmetry paradox in a later section. 29 30 Continuous Shape and Symmetry Measures 31 32 33 According to the proposal of Avnir and coworkers,[7, 16] in order to obtain a shape measure 34 35 for a structure X (represented in 1 by the circles joined by dashed lines) we need first to search 36 for the ideal shape A (represented in 1 by a ) that is closest to our problem structure. This 37 38 search requires optimization with respect to size, orientation in space and pairing of vertices of 39 the two structures. Once the reference shape is found, we calculate the distances between the 40 41 equivalent atomic positions in the two structures, qk, from which we calculate the shape measure 42 according to equation 1, where N is a normalization factor that makes the continuous shape 43 44 measures (CShM) values size independent. To optimize A, SX(A) must be minimized with 45 respect to size, orientation and vertex pairing. 46 47 48 49 50 51 1 52 53 54 55 N 56 2 57 !qk k =1 58 SX (A) = min 100 [1] 59 N 60

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1 2 3 4 From the definition of equation 1 it can be shown that the SX(A) values must lie between 0 5 and 100. The resulting value is zero if the problem structure X has exactly the desired shape A, 6 7 and will increase with the degree of distortion. As a rule of thumb we can say that chemically 8 significant distortions should give CShM values higher than about 0.1, while values of the order 9 [17] 10 of 2 or higher indicate important distortions. Since all CShM values are in the same scale, 11 independently of the reference shape adopted and the number of vertices, we can compare, for 12 13 instance, the deviation of a given structure from different reference shapes or of different 14 structures with respect to the same ideal shape. 15 16 17 Some Polyhedral Alkanes 18 19 20 Since the fullerenesFor form Peer a wide family Review and a huge number of publications have been 21 devoted to them, we concentrate here on the more sparse examples of polyhedral molecules 22 23 from other families. Some structurally characterized polyhedral alkanes are summarized in 24 Table 1, most of them with Platonic or Archimedean polyhedral shapes. The Johnson and 25 26 Catalan polyhedra all have vertices with connectivity four or higher, making them poorly 27 adapted for cage alkanes. Nevertheless, if we allow some of the edges of a polyhedron to 28 29 correspond to non-bonded atoms, accepting some deviation from the , we may 30 find a couple of Johnson polyhedranes. One of them is bicyclo(1.1.1)pentane, which resembles 31 32 a trigonal , with structures of its derivatives presenting shape measures relative to such 33 34 a polyhedron between 1 and 2. Cunneanes (Figure 2), in contrast, deviate significantly from the 35 ideal gyrobifastigium (shape measures of about 8.8), due to the presence of two non-bonded 36 37 edges. However, the gyrobifastigium is still the polyhedral shape that best describes the 38 structures of cunneanes. Another interesting irregular polyhedron is that showcased by 39 40 octahedrane (Figure 2), sometimes named "melancholyhedron" because it first appeared in an 41 well known engraving from Albrecht Dürer titled Melancholy.[6] 42 43 44 45 46 47 48 Trigonal Bipyramid Gyrobifastigium Melancholyhedron 49 50 51 52 53 54 55 56 57 58 59 Figure 2. Structures of the carbon skeletons of bicyclopentane, cunneane and octahedrane, that 60 approximate the of the trigonal bipyramid, the gyrobifastigium, and the melancholyhedron, respectively.

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1 2 3 4 Table 1. Structurally characterized polyhedranes. 5 6 7 Compound n Polyhedron Family Ref. 8 9 Experimental Structures 10 11 Tetrahedrane 4 Tetrahedron Platonic [18]

12 [19] 13 Bicyclo(1.1.1)pentane 5 ~ Trigonal Bipyramid Johnson 14 [20] 15 6 Trigonal Prism 16 Cubane 8 Cube Platonic [21] 17 18 Cunneane 8 ~ Gyrobifastigium Johnson [22] 19 20 Pentaprismane For10 PeerPentagonal Review Prism Prism [23] 21 [24] 22 Hexaprismane 12 Prisms 23 [25] 24 Octahedrane 12 Melancholyhedron Irregular 25 Dodecahedrane 20 Dodecahedron Platonic [26] 26 27 Theoretical Structures 28 29 Truncoctahedrane 24 Truncated Octahedron Archimedean [27] 30 31 Fullerane 60 Truncated Icosahedron Archimedean [28] 32 33 34 For a polyhedron whose vertices are made of CH groups to be feasible, it must have only 35 three-connected vertices. In the corresponding polyhedrane, the three edges meeting at each 36 37 vertex corespond to C-C bonds. There are three Platonic and seven Archimedean polyhedra that 38 comply with this requirement, two of which have been reported only as theoretical molecules 39 40 (Table 1), although they appear scattered in the literature. To have a full gallery of all possible 41 Platonic and Archimedean polyhedranes, we have optimized them via DFT calculations (Figure 42 43 3) and found them to correspond to minima in the corresponding potential energy surfaces, with 44 carbon-carbon bond distances characteristic of single bonds. It is customary to relate the 45 46 energies of polyhedral molecules to the deviation of the carbon atoms from the tetrahedral 47 48 geometry imposed by the shape of the polyhedron. Thus, a energy is defined, which is 49 calculated as the total energy divided by the number of CH groups, taking as zero strain energy 50 51 the value obtained for the most tetrahedral case, which corresponds to dodecahedrane (equation 52 2). The resulting strain energies show the expected dependence on the H-C-C bond angles, and 53 54 the most stable polyhedra are those that present average H-C-C bond angles closer to the 55 tetrahedral angle (Table 2). 56

57 Etotal (CnHn polyhedrane) Etotal (C20H20 dodecahedrane) 58 Es (CnHn polyhedrane) = ! [2] 59 n 20 60

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1 2 3 4 5 Table 2. Strain energies (Es, kcal/mol) and bonding parameters for several optimized 6 polyhedranes with general formula CnHn. All distances in Å, angles in degrees; experimental 7 values given in parentheses when available. 8 9 [a] 10 n Polyhedron Es C-C (Å) H-C-C (º) Σ (º) 11 12 120 Trunc. Icosidodecahedron 23.4 1.58-1.61 101.7 354 13 96.7 14 15 60 25.4 1.545 101.7 348 16 17 1.534 97.5 18 60 Truncated Icosahedron 7.4 1.571 101.6 345 19 20 For Peer Review1.557 21 22 48 13.0 1.579 102.8 345 23 1.546 24 25 24 13.2 1.525 108.0 330 26 1.513 27 28 24 Truncated Octahedron 5.8 1.569 109.2 330 29 30 1.533 107.2 31 20 Dodecahedron 0.0 1.556 (1.545) 110.9 (111) 324 32 33 12 6.7 1.522 116.1 300 34 35 1.499 114.2 36 37 12 Hexagonal prism 11.3 1.560 115.1 300 38 (hexaprismane) 1.566 122.1 39 40 10 Pentagonal prism 10.7 1.561 (1.55) 119.4 (119) 288 41 (pentaprismane) 1.570 (1.57) 123.4 (124) 42 43 8 Cube 16.7 1.571 (1.551) 125.3 (126) 270 44 45 6 Trigonal Prism (prismane) 20.8 1.522 (1.53) 129.6 (130) 240 46 1.558 (1.55) 132.7 (132) 47 48 4 Tetrahedron 31.1 1.479 (1.486) 144.7 (145) 180 49 50 51 [a] Sum of C-C-C bond angles around a carbon atom. 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 6 7 8 9 10 11 12 13 C12H12 C24H24 C24H24 C48H48 14 15 Truncated Tetrahedron Truncated Cube Truncated Octahedron Truncated Cuboctahedron 16 17 18 19 20 For Peer Review 21 22 23 24 25 26 27 C60H60 C60H60 C120H120 28 Truncated Dodecahedron Truncated Icosahedron Truncated Icosidodecahedron 29 30 Figure 3. Calculated structures of the Archimedean alkanes. 31 32 33 The of some Archimedean polyhedra are not well suited for fully 34 dehydrogenated C molecules, because of the non-planarity of their vertices (measured by the 35 n 2 36 sum of the subtended C-C-C bond angles, Σ, that should be 360º for an ideal sp carbon atom, 37 see Table 2). Nevertheless, small deviations from planarity can be withstood, as in the truncated 38 39 icosahedron of C60. Theoretically optimized structues of other Archimedean polyhedrenes have 40 been reported in separate studies (Table 3), to which we can add the truncated cube. It must be 41 42 noted that different delocalization schemes may appear in these molecules, as evidenced for 43 [29] 44 several alternative structures of the Platonic dodecahedrene C20. 45 46 Table 3. Reported theoretical studies of Archimedean and Platonic polyhedrenes with general 47 formula C . 48 n 49 50 n Polyhedron Ref. 51 [30] 52 120 Truncated Icosidodecahedron 53 60 Truncated Icosahedron [31] 54 55 48 Truncated Cuboctahedron [32] 56 57 24 Truncated Cube this work 58 59 24 Truncated Octahedron [33] 60 12 Truncated Tetrahedron [34]

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1 2 3 4 Shape Maps and Interconversion Pathways 5 6 For the stereochemical analysis of families of compounds we have found it useful to 7 8 represent scatterplots of their shape measures relative to two alternative ideal polyhedra with the 9 same number of vertices (e.g., A and B), that we call shape maps. In these maps, the lower left 10 11 limit always corresponds to the interconversion path between the two reference shapes, usually 12 polyhedra or polygons. The shape measures of all structures i along the minimal distortion 13 14 interconversion path between polyhedra A and B, Si(A) and Si(B), must obey the following 15 [9] 16 relationship: 17

18 Si (A) Si (B) 19 arcsin + arcsin = ! [3] 10 10 AB 20 For Peer Review 21 22 [9] where θAB is a constant for each pair of polyhedra, the symmetry angle. Structures that do not 23 24 belong to the minimal distortion path do not obey equation 3 and their distance to that path can 25 be calibrated by means of the path deviation function defined in equation 4, where x refers to an 26 27 arbitrary structure. 28 29 1 # S (A) S (B) & 30 x x ! x (A, B) = %arcsin + arcsin ( ) 1 [4] 31 "AB $% 10 10 '( 32 33 34 Furthermore, for structures that are along the minimal interconversion pathway we have defined 35 a generalized interconversion coordinate,[35] that measures the percentage of the path between 36 37 two polyhedra covered by the problem structure (equation 5). 38 39 100 $ S (B) ' 40 ! = arcsin i [5] 41 A" B & ) #AB % 10 ( 42 43 44 45 With such tools we can (a) detect very easily those structures that are intermediate 46 between two ideal shapes, (b) obtain a quantitative description of how close (or how far) a given 47 48 structure is from a path, (c) obtain molecular models of the shapes that correspond to steps along 49 the interconversion path, and (d) calibrate the distance of the problem structure to the two 50 51 extremes of the path. According to the proposal that data offer clues to reaction 52 pathways,[36] the analysis of crystal structures from the point of view of minimal distortion paths 53 54 should be helpful in gaining insight into chemical reactivity aspects. 55 56 To illustrate the possible applications of shape maps, we show in Figure 4 the structural 57 58 data of all non-fused cyclohexyl groups found in the Cambridge Database in two different shape 59 60 maps. In the first one, the structures are compared with the planar and the octahedron, since its chair conformation can be seen as an intermediate between those two ideal shapes

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1 2 3 4 (Scheme I). We have found that 97% of the cyclohexyl groups found are aligned along the 5 minimal distortion path between the hexagon and the octahedron, with a deviation of at most 6 7 10% (the small portion of molecules that deviate most from that path are not shown in the plot 8 for clarity). 9 10 11 An alternative way of looking at the data plotted in Figure 4 consists in analyzing the 12 frequency of structures found at different steps along the path, shown in the histogram of Figure 13 14 5. There we see that the structures are strongly concentrated around a generalized coordinate 15 16 ϕOC→HP of 74%, suggesting that such a conformation is the most stable one. In effect, a DFT 17 optimization of the molecular structure of cyclohexane in the chair conformation appears at a 18 19 generalized coordinate ϕOC→HP of 74.8%, exactly on the track from the octahedron to the 20 planar hexagon. For Peer Review 21 22 Scheme I 23 24 25 26 27 28 29 Hexagon Chair Octahedron 30 31 32 33 34 35 36 Hexagon Boat Trigonal Prism 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Figure 4. Shape maps for the transformations of a hexagon into an octahedron (left) and into a 57 trigonal prism (right). Structural data for non-fused cyclohexyl groups with deviations from the 58 paths larger than 10% are omitted for clarity. Structures of prismanes also shown in the 59 60 hexagon-trigonal prism map for comparison.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Figure 5. FrequencyFor of cyclohexyl Peer groups Review as a function of their degree of hexagonality 21 (generalized coordinate for the octahedron-planar hexagon conversion), in a logarithmic scale. 22 23 24 We figured out that the cyclohexyl groups that significantly deviate from this path should 25 probably present a boat conformation. Therefore, we have plotted those structures in a shape 26 27 map relative to the hexagon and the trigonal prism, since the boat structures are in-between 28 29 those two ideal shapes. The results (Figure 4, right) clearly discriminate the boat from the chair 30 structures, but also allow us to detect different degrees of each of those two conformations. Of 31 32 course, it is not possible to ascribe in an unequivocal way a boat or chair conformation to those 33 molecules that are close to the planar hexagon. 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Figure 6. Structural data of cyclooctane, cyclooctatetraene (COT) and cubane derivatives, 52 plotted in a shape map relative to the cube (CU-8) and the planar (OP-8). The line 53 54 represents the minimum distortion interconversion path between the octagon and the cube. 55 56 57 58 59 60

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1 2 3 Scheme II 4 Cp 5 6 Rh 7 OC CO 8 O 9 C + 2- 10 Ru Ru (K )2 11 OC CO 12 Rh 13 14 Cp 15 S(Octagon): 38.2 12.3 7.7 5.2 2.4 0.0 16 17 S(Cube): 0.0 14.2 20.4 24.2 30.2 38.1 18 19 20 A second case thatFor we have Peer analyzed is thatReview of the cyclooctatetraene groups (COT). Those 21 skeletons are expected to be in a boat shape (reminiscent of the cube) for the neutral COT, but 22 23 perfectly octagonal for the aromatic dianion. It is not surprising, therefore, that the structures of 24 all C R groups found in the CSD are aligned to a good approximation along the coresponding 25 8 8 26 minimal distortion interconversion path (Figure 6). It is interesting to note that there is no 27 geometric discontinuity between the aromatic dianionic rings and the neutral tetraenes. In 28 29 addition, the absence of structures in a wide portion of the pathway is suggestive of a high 30 energy barrier for the interconversion of the two , cyclooctatetraene and cubane, through 31 32 the pathway analyzed here. Let us finally stress that cyclooctatetraene, C8H8, is not among the 33 systems that most deviate from planarity. Some of the compounds that appear along the path are 34 35 shown in Scheme II, together with their shape measures. 36 37 38 Finally, a look at the structures of cyclobutane derivatives (disregarding those with fused 39 rings) shows also that their conformations can be nicely described as being along the square to 40 41 tetrahedron pathway (Scheme III). Among 551 crystallographically independent data sets 42 found, all cyclobutane skeletons deviate less than 5% from that path with only two exceptions, 43 44 covering the range from strictly planar to 28% bending toward the tetrahedron. 45 46 Scheme III 47 48 49 50 51 Square Tetrahedron 52 53 54 55 Symmetry Paradox 56 57 We have recently shown that octahedral symmetry is latent in polyhedra of icosahedral 58 [14] 59 symmetry, even if the corresponding symmetry point groups are incompatible. A 60 consequence of such a relationship is the capability of molecules with to form a network of intermolecular interactions with octahedral symmetry that results quite often

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1 2 3 4 in cubic crystal structures of icosahedrally symmetric molecules. In Table 4 we present some 5 examples of organic polyhedral molecules that form cubic . 6 7 8 Table 4. Cubic crystal structures of molecules with icosahedrally symmetric carbon polyhedra. 9 Compd. Packing[a] Polyhedron Ref. 10 [37] 11 C20H20(dodecahedrane) fcc Dodecahedron 12 C () fcc Truncated Icosahedron [38] 13 60 [39] 14 C60·H2C=CH2 fcc Truncated Icosahedron 15 K Ba C bcc Truncated Icosahedron [40] 16 3 3 60 [41] 17 C60·O2 fcc Truncated Icosahedron 18 19 [a] fcc = -centered cubic; bcc = body-centered cubic. 20 For Peer Review 21 We think that octahedrally-arranged atoms or functional groups can also be attached to 22 23 icosahedral molecules. We have noticed, for instance, that buckminsterfullerene can be 24 25 considered as formed by six units whose centers are arranged in an octahedral way. 26 Alternatively, it can also be described as an assembly of six units, whose centers 27 28 occupy the vertices of an octahedron (Figure 7). Chemical substitutions that occupy the centers 29 of those bonds would result in an octahedron circumscribed around the C60 truncated 30 31 icosahedron. As an example, we have computationally optimized the structure of a hexa- 32 epoxidized fullerene, in which the six oxygen atoms are seen to form a perfect octahedron 33 [42] 34 (Figure 8). In a related experimental structure, the same positions are occupied by 35 cyclopropane rings, whose carbon atoms also form a perfect octahedron. In both cases, the six 36 37 substituents directly attached to the fullerene have octahedral shape measures smaller than 0.01, 38 while the fullerene core deviates very little from its truncated icosahedral shape (shape measures 39 40 of 0.03 and 0.06 for the epoxidized and cyclopropanated derivatives, respectively). 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 Figure 7. Buckminsterfullerene C60 as an assembly of six naphthalene units (left) or six 59 60 fulvalene units (right).

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 Figure 8. Left: Epoxidized fullerene whose six oxygen atoms form a perfect octahedron, 23 templated by the C60 truncated icosahedron (computational DFT results). Right: Analogous 24 [42] 25 octahedron formed by six cyclopropane fused rings in an experimentally reported compound. 26 27 Another computational example of a molecule that combines the icosahedral symmetry of 28 29 a dodecahedrane skeleton with the cubic symmetry of eight substituents is that of C20H12Br8, 30 shown in Figure 9. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Figure 9. Optimized structure of an octabrominated dodecahedrane C H Br with a 46 20 12 8 47 substitution pattern that displays a cubic set of atoms ( atoms not shown for 48 clarity) circumscribed around the dodecahedron of carbon atoms. 49 50 51 Although is not usually perceived as a polyhedral molecule, it can be 52 described as a composite of a tetrahedron of tertiary carbon atoms (2) bridged by secondary 53 [6] [43] 54 carbon atoms that form a circumscribed octahedron. Omar Yaghi and coworkers have taken 55 advantage of the tetrahedron implicit in adamantane to design a network of a porous metal 56 57 organic framework (MOF) reminiscent of the PtS structure but with larger voids. The 58 tetrahedral adamanane unit is provided with four carboxylate groups that have the same spatial 59 60 connctivity than a sulfide ion. Those carboxylates are coordinated to a pair of copper(II) ions

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1 2 3 II 4 forming a square four-connected unit (3), that plays the same role of the Pt ion in PtS. The 5 analogous construction principle of the two structures can be appreciated in Figure 10. 6 7 8 9 10 11 12 13 14 15 16 2 3 17 18 19 20 For Peer Review 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Figure 10. Formal replacement of the tetrahedral sulfide ions in PtS by adamantane units (2) II 41 and of the square planar Pt ions by Cu2(carboxylate)4 building blocks (3). 42 43 44 Concluding Remarks 45 46 An assortment of beautifully symmetrical polyhedra can be expressed as organic 47 48 molecules, and several examples have been obtained in the bench or in the computer. We have 49 shown here also that thinking in terms of polyhedral shapes may be helpful to analyze the 50 51 stereochemistry of non polyhedral molecules. The implicit polyhedra found in adamantane, for 52 instance, has beeen used by Yagi and coworkers to design and build up porous extended 53 54 frameworks. The latent cubic symmetry present in molecues with icosahedral symetry is 55 important in establishing the directions of intermolecular interactions that often result in crystals 56 57 with cubic packings, as in dodecahedrane and buckminsterfullerene. 58 59 60

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1 2 3 Computational Details 4 5 [44] 6 DFT calculations were carried out with Gaussian03, using the B3LYP hybrid functional 7 and a 6-31G** Gaussian basis set.[45] All calculated structures reported were characterized as 8 9 true minima through vibrational analysis. Structural searches were carried out in the CSD,[46] 10 [47] 11 version 5.30 with three updates, and the atomic coordinates transferred to the SHAPE code to 12 calculate continuous shape measures, path deviation functions and generalized coordinates.[48] 13 14 15 Acknowledgments 16 17 This work has been supported by the Ministerio de Invesigación, Ciencia e Innovación 18 19 (MICINN), project CTQ2008-06670-C02-01-BQU, and by Generalitat de Catalunya, grants 20 2009SGR-1459 and XRQTC.For Allocation Peer of computer Review time at the Centre de Supercomputació de 21 22 Catalunya, CESCA, is greatfully acknowledged. J. E. thanks the Spanish Ministerio de 23 Educación for an FPU grant (reference AP2008-02735). 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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